In general, the value function associated with an exit time problem is a discontinuous function. We prove that the lower (upper) semicontinuous envelope of the value function is a supersolution (subsolution) of the Hamilton–Jacobi equation involving the proximal subdifferentials (superdifferentials) with subdifferential-type (superdifferential-type) mixed boundary condition. We also show that i...

We study constrained viscosity solutions with an unbounded growth for a class of first order Hamilton–Jacobi–Bellman equations arising in hybrid control systems. To deal with the boundary constraint and rapid growth of the solutions, we construct a particular set of test functions and under very mild conditions establish a comparison theorem which gives the estimate of distance between the subs...

We consider a robust switching control problem. The controller only observes the evolution of the state process, and thus uses feedback (closed-loop) switching strategies, a non standard class of switching controls introduced in this paper. The adverse player (nature) chooses open-loop controls that represent the so-called Knightian uncertainty, i.e., misspecifications of the model. The (half) ...

Numerical solutions of 2× 2 semilinear systems of elliptic boundary value problems, whose nonlinearities are of quasimonotone nondecreasing, quasimonotone nonincreasing, or mixed quasimonotone types, are computed. At each step of the (quasi) monotone iteration, the solution is represented by a simple-layer potential plus a domain integral; the simple-layer density is then discretized by boundar...

We formulate a robust optimal control problem for a general nonlinear system with finitely many admissible control settings and with costs assigned to switching of controls. We formulate the problem both in an L2-gain/dissipative system framework and in a game-theoretic framework. We show that, under appropriate assumptions, a continuous switching-storage function is characterized as a viscosit...

In this paper we prove a sufficient condition, in terms of the behavior of a ground state of a symmetric critical operator P1, such that a nonzero subsolution of a symmetric nonnegative operator P0 is a ground state. Particularly, if Pj := −∆ + Vj , for j = 0, 1, are two nonnegative Schrödinger operators defined on Ω ⊆ R such that P1 is critical in Ω with a ground state φ, the function ψ 0 is a...

We consider the use of Lagrange manifolds to construct viscosity solutions of first order Hamiltonian-Jacobi equations. Recent work of several authors is indicated in which the essential underlying structure consists of a Lagrange manifold on which 1) the desired Hamiltonian function vanishes and 2) the canonical 1-form p·dx of classical mechanics has an integral S(x, p). We explore the proposi...

Firstly, we define an order for differential forms. Secondly, we also define the supersolution and subsolution of the A-harmonic equation and the obstacle problems for differential forms which satisfy theA-harmonic equation, and we obtain the relations between the solutions toA-harmonic equation and the solution to the obstacle problem of the A-harmonic equation. Finally, as an application of t...

Half space problems of the linear Boltzmann equation with a constant driving force are considered. Such problems model boundary layers between kinetic zones and fluid zones described by a high field limit of the Boltzmann equation. Existence, uniqueness, and asymptotic behavior of solutions are studied for positive and negative driving forces. In the positive case, the force field accelerates t...

In the present paper we prove a multiplicity theorem for a quasilinear elliptic problem with dependence on the gradient ensuring the existence of a positive solution and of a negative solution. In addition, we show the existence of the extremal constant-sign solutions: the smallest positive solution and the biggest negative solution. Our approach relies on extremal solutions for an auxiliary pa...